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Chapter 11: Problem 17

Evaluate each of the numerical expressions. $$64^{3}$$

Short Answer

Expert verified
262144

Step by step solution

01

Understanding the Expression

The expression given is \(64^3\), which means 64 multiplied by itself three times. So, we need to calculate \(64 \times 64 \times 64\).
02

Calculate 64 Squared

First, calculate \(64^2\). Multiply 64 by itself: \(64 \times 64 = 4096\). Now we have the square of 64.
03

Multiply by 64 Again

Take the result from the previous step, 4096, and multiply it by 64 to find \(64^3\): \[4096 \times 64 = 262144\]
04

Conclusion

The value of \(64^3\) is 262144.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Numerical Expressions
Numerical expressions are like sentences but with numbers instead of words. A numerical expression represents a particular value. In the exercise given, the expression is represented by exponentiation, specifically the power of a number, which is a very common form of numerical expression. When you see an expression like \(64^3\), it means 64 is raised to the power of 3, representing a multiplication process. It's important to read and understand expressions as they guide you on how to calculate the result step by step. Parsing them accurately can help make the calculations easier to handle.
Breaking Down the Multiplication Process
The multiplication process is a pivotal part of working with exponents. Let's break it down using \(64^3\) as an example. Here, multiplication plays an essential role in calculating powers of numbers. Begin by multiplying the base number by itself repeatedly. For \(64^3\), this means you start with \(64 \times 64\) to get 64 squared.
  • First, calculate \(64 \times 64\), which equals 4096.
  • Then, use that result and multiply it by 64 again, giving you \(4096 \times 64 = 262144\).
Through each multiplication step, you get closer to finding the final value. Remember, creating smaller, manageable steps in calculations can simplify the problem. Repeated multiplication through the multiplication process is often necessary when dealing with higher powers.
Understanding Powers of Numbers
Powers of numbers simplify complex multiplication. When a number is raised to a power, such as \(64^3\), it tells you how many times to multiply the base number (64) by itself. This concept is vital for simplifying long expressions. Breaking the power down:
  • Exponent \(3\) indicates multiplying 64 three times: \(64 \times 64 \times 64\).
  • The base can be any number, and the exponent tells how many times the base acts as a factor.
Understanding powers helps handle big numbers effectively by reducing a long multiplication sequence into a streamlined calculation. The use of exponents is popular in various fields, such as science and finance, where large or repeated calculations are common.

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Most popular questions from this chapter

For Problems \(35-54\), find each product and express it in the standard form of a complex number \((a+b i)\). $$ (4+5 i)^{2} $$

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